Theorem hoidmvval 40791

Description: The dimensional volume of a multidimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
hoidmvval.l	|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )
hoidmvval.a	|- ( ph -> A : X --> RR )
hoidmvval.b	|- ( ph -> B : X --> RR )
hoidmvval.x	|- ( ph -> X e. Fin )

Assertion

Ref	Expression
hoidmvval	|- ( ph -> ( A ( L ` X ) B ) = if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) )

Distinct variable groups:   x, k   A, a, b, k   B, a, b, k   X, a, b, k, x   ph, a, b, x

Allowed substitution hints:   ph( k)   A( x)   B( x)   L( x, k, a, b)

Proof of Theorem hoidmvval

Step	Hyp	Ref	Expression
1		hoidmvval.l	. . . 4 |- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )
2	1	a1i 11	. . 3 |- ( ph -> L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) ) )
3		oveq2 6658	. . . . 5 |- ( x = X -> ( RR ^m x ) = ( RR ^m X ) )
4		eqeq1 2626	. . . . . 6 |- ( x = X -> ( x = (/) <-> X = (/) ) )
5		prodeq1 14639	. . . . . 6 |- ( x = X -> prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) = prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) )
6	4, 5	ifbieq2d 4111	. . . . 5 |- ( x = X -> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) = if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) )
7	3, 3, 6	mpt2eq123dv 6717	. . . 4 |- ( x = X -> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) = ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )
8	7	adantl 482	. . 3 |- ( ( ph /\ x = X ) -> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) = ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )
9		hoidmvval.x	. . 3 |- ( ph -> X e. Fin )
10		ovex 6678	. . . . 5 |- ( RR ^m X ) e. _V
11	10, 10	mpt2ex 7247	. . . 4 |- ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) e. _V
12	11	a1i 11	. . 3 |- ( ph -> ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) e. _V )
13	2, 8, 9, 12	fvmptd 6288	. 2 |- ( ph -> ( L ` X ) = ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )
14		fveq1 6190	. . . . . . . 8 |- ( a = A -> ( a ` k ) = ( A ` k ) )
15	14	adantr 481	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( a ` k ) = ( A ` k ) )
16		fveq1 6190	. . . . . . . 8 |- ( b = B -> ( b ` k ) = ( B ` k ) )
17	16	adantl 482	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( b ` k ) = ( B ` k ) )
18	15, 17	oveq12d 6668	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( ( a ` k ) [,) ( b ` k ) ) = ( ( A ` k ) [,) ( B ` k ) ) )
19	18	fveq2d 6195	. . . . 5 |- ( ( a = A /\ b = B ) -> ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) = ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) )
20	19	prodeq2ad 39824	. . . 4 |- ( ( a = A /\ b = B ) -> prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) )
21	20	ifeq2d 4105	. . 3 |- ( ( a = A /\ b = B ) -> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) = if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) )
22	21	adantl 482	. 2 |- ( ( ph /\ ( a = A /\ b = B ) ) -> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) = if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) )
23		hoidmvval.a	. . 3 |- ( ph -> A : X --> RR )
24		reex 10027	. . . . 5 |- RR e. _V
25	24	a1i 11	. . . 4 |- ( ph -> RR e. _V )
26		elmapg 7870	. . . 4 |- ( ( RR e. _V /\ X e. Fin ) -> ( A e. ( RR ^m X ) <-> A : X --> RR ) )
27	25, 9, 26	syl2anc 693	. . 3 |- ( ph -> ( A e. ( RR ^m X ) <-> A : X --> RR ) )
28	23, 27	mpbird 247	. 2 |- ( ph -> A e. ( RR ^m X ) )
29		hoidmvval.b	. . 3 |- ( ph -> B : X --> RR )
30		elmapg 7870	. . . 4 |- ( ( RR e. _V /\ X e. Fin ) -> ( B e. ( RR ^m X ) <-> B : X --> RR ) )
31	25, 9, 30	syl2anc 693	. . 3 |- ( ph -> ( B e. ( RR ^m X ) <-> B : X --> RR ) )
32	29, 31	mpbird 247	. 2 |- ( ph -> B e. ( RR ^m X ) )
33		c0ex 10034	. . . 4 |- 0 e. _V
34		prodex 14637	. . . 4 |- prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. _V
35	33, 34	ifex 4156	. . 3 |- if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) e. _V
36	35	a1i 11	. 2 |- ( ph -> if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) e. _V )
37	13, 22, 28, 32, 36	ovmpt2d 6788	1 |- ( ph -> ( A ( L ` X ) B ) = if ( X = (/) , 0 , prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   RRcr 9935   0cc0 9936   [,)cico 12177   prod_cprod 14635   volcvol 23232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-prod 14636

This theorem is referenced by:  hoidmvcl  40796  hoidmv0val  40797  hoidmvn0val  40798  hsphoidmvle  40800